There are 3 servers in the checkout area. The interarrival time of customers is 2 minutes. The processing time is 5 minutes. The coefficients of variation for the arrival process and the service process are 1 and 0.85 respectively. What will be the impact on the average number of customers in service if one more server is added

As the one more server is added the Average number of customers in service reduces from  ( 5.585 to 3.019  )    which is a reduction of 2.566

Explanation:

interval time of customers ( a )  = 2 minutes

processing time ( p ) = 5 minutes

coefficient of variation for arrival process ( Cva ) = 1

coefficient of variation for service process ( Cvp ) = 0.85

Determine the impact on the average number of customers in service if one more server is added

For three (3) servers (  m ) we will have

flow rate = 1 / a = 1/2 = 0.5

x =  ( 2 (m+1) -1 ) ^0.5

where m = 3 ( number of servers )

hence X = 2.645

now lets determine

i) utilization =  p / (m*a). where p = 5 , m = 3 , a = 2

hence utilization ( u ) = 0.833 ,  1 - u = 1 - 0.833 = 0.167

ii) Time spent in queue

time spent in queue = 6.17 minutes

number of customers been served = mu = 3 * 0.833 = 2.500

number of customers waiting to be served = waiting time / arrival time

= 6.17 / 2 = 3.085

iii) Average customer in queue = number of customers been serviced + number of customers waiting to be served =  2.5 + 3.085 = 5.585

For 4 servers  ( m = 4 )

x = ( 2 (m+1) -1 ) ^0.5  

   = ( 2 ( 5 ) - 1 ) ^ 0.5 = 3

utilization =  p / (m*a). where p = 5 , m = 4 , a = 2

   hence utilization = 0.625 ,  

Time spent in queue =  1.039 minutes  ( using equation 2 )

number of customers been serviced = m* u = 2.5

number of customers waiting to be served = waiting time / arrival time

= 1.039 / 2 = 0.520

Average customer in queue = number of customers been serviced + number of customers waiting to be served = 2.5 + 0.520 = 3.019

hence as the one more server is added the Average number of customers in service reduces from  ( 5.585 to 3.019  )    which is a reduction of 2.566

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explanation:

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